Spin connection

I want to know wether the component [itex]\omega_{c}^{ab}[itex] of the spin connection is zero or not?

2. Relevant equations
[itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[itex]

3. The attempt at a solution
[itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[itex]
##=-\omega^{a}_{b,\nu} e^{b}_{\mu} dx^{\nu}\wedge dx^{\mu}##
##=-\omega^{a}_{b,c} e^{b}_{[\mu}e^{c}_{\nu]} dx^{\nu}\wedge dx^{\mu}##
as it is antisymmetric in ##\mu## and ##\nu##.So it is also antisymmetric in b and c.Thus one can conclude from here.

I want to know wether the component [itex]\omega_{c}^{ab}[/itex] of the spin connection is zero or not?

2. Relevant equations
[itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[/itex]

3. The attempt at a solution
[itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[/itex]
##=-\omega^{a}_{b,\nu} e^{b}_{\mu} dx^{\nu}\wedge dx^{\mu}##
##=-\omega^{a}_{b,c} e^{b}_{[\mu}e^{c}_{\nu]} dx^{\nu}\wedge dx^{\mu}##
as it is antisymmetric in ##\mu## and ##\nu##.So it is also antisymmetric in b and c.Thus one can conclude from here.

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